Vibration isolation is the engineering practice of reducing the transfer of oscillatory motion or force between a machine and its support. This cheat sheet helps students connect spring, mass, and damper models to practical isolation design. It is useful for analyzing rotating equipment, precision instruments, vehicle mounts, and building mechanical systems.
The main goal is to choose stiffness and damping so harmful vibration is reduced over the required frequency range.
The core model is a single-degree-of-freedom mass-spring-damper system driven by either a harmonic force or moving base. The most important quantities are natural frequency, damping ratio, frequency ratio, transmissibility, and isolation efficiency. Isolation usually begins when the excitation frequency is greater than about sqrt(2) times the natural frequency.
Damping reduces resonance peaks but can reduce high-frequency isolation performance.
Key Facts
- The undamped natural circular frequency is omega_n = sqrt(k/m), where k is stiffness and m is supported mass.
- The natural frequency in hertz is f_n = (1 / 2 pi) sqrt(k/m).
- The damping ratio is zeta = c / c_c, where c_c = 2 sqrt(km) is the critical damping coefficient.
- The frequency ratio is r = omega / omega_n = f / f_n, where omega or f is the excitation frequency.
- For harmonic force isolation, force transmissibility is T = sqrt(1 + (2 zeta r)^2) / sqrt((1 - r^2)^2 + (2 zeta r)^2).
- Isolation occurs when T < 1, which for light damping begins approximately when r > sqrt(2).
- Isolation efficiency is eta = (1 - T) x 100 percent when T is less than 1.
- Static deflection under weight is delta = mg/k, and the vertical natural frequency can be estimated by f_n = (1 / 2 pi) sqrt(g/delta).
Vocabulary
- Transmissibility
- Transmissibility is the ratio of transmitted vibration force or motion to the input force or motion.
- Natural Frequency
- Natural frequency is the frequency at which a system tends to vibrate freely after being disturbed.
- Damping Ratio
- Damping ratio is a nondimensional measure of damping compared with critical damping.
- Frequency Ratio
- Frequency ratio is the excitation frequency divided by the system natural frequency.
- Static Deflection
- Static deflection is the steady displacement of an isolator caused by the supported weight.
- Resonance
- Resonance is the condition in which excitation frequency is near natural frequency and vibration amplitude can become large.
Common Mistakes to Avoid
- Using hertz and radians per second interchangeably is wrong because omega = 2 pi f, so missing the 2 pi factor changes natural frequency and transmissibility.
- Assuming more damping always improves isolation is wrong because damping lowers the resonance peak but increases transmitted motion at high frequency ratios.
- Designing for r near 1 is wrong because this places the system near resonance, where amplitudes and transmitted forces can be much larger than the input.
- Ignoring static deflection is wrong because a low natural frequency usually requires a soft isolator, which may sag too much or exceed clearance limits.
- Applying the force transmissibility formula to every base-motion problem is wrong because base excitation can require displacement transmissibility and relative motion checks.
Practice Questions
- 1 A 120 kg machine is mounted on isolators with total stiffness k = 48,000 N/m. Find omega_n and f_n.
- 2 An isolator has f_n = 6 Hz and the machine excitation frequency is 24 Hz. Find the frequency ratio r and state whether isolation is expected.
- 3 For zeta = 0.10 and r = 3, calculate the force transmissibility T using T = sqrt(1 + (2 zeta r)^2) / sqrt((1 - r^2)^2 + (2 zeta r)^2), then find isolation efficiency.
- 4 Explain why an isolator with a very soft spring can reduce transmitted vibration but still be unacceptable in a real machine installation.
Understanding Vibration Isolation Reference
Two loading cases lead to different design checks. In force excitation, a machine creates a periodic force within its own frame. A motor with a slightly unbalanced rotor is a common example.
The isolator must limit the force reaching the floor or supporting structure. In base excitation, the support itself moves. A vehicle seat, an instrument on a vibrating platform, or equipment in an earthquake zone can face this condition.
The quantity of interest is often the motion of the supported mass relative to the moving base. The same mass, spring, and damper model applies, but the input and output are different. Students should identify the input before selecting a transmissibility equation.
Resonance is the difficult region near the system natural frequency. Here, even a small repeated input can produce large motion. The stored energy moves back and forth between the mass and spring.
If damping is low, this exchange continues for many cycles and the response can become large. Large relative motion can cause a spring to bottom out, a mount to tear, or attached pipes and cables to fatigue. A design therefore needs enough travel space around the machine.
It must survive start-up and shut-down too. A rotating machine passes through a range of speeds during these events, so it may briefly operate near resonance even if its normal running speed is safely above it.
Damping creates an important compromise. Adding damping lowers the resonance peak, which protects equipment during speed changes and reduces excessive motion. At frequencies far above the natural frequency, however, heavy damping provides a stronger path for vibration to cross the mount.
Rubber mounts have inherent damping, while viscous dashpots use fluid resistance. Some systems use separate restraints or snubbers that act only when motion becomes too large.
The best damping ratio depends on the operating frequency range, expected shocks, allowable movement, and consequences of resonance. A mount that looks effective during a low-speed test may perform poorly at the machine's actual operating speed.
Static deflection gives a useful physical check on the calculation. A softer vertical support settles farther under the equipment weight and usually has a lower natural frequency. This is why heavy machinery may sit on thick rubber pads, steel springs, or air mounts.
Softness cannot be increased without limit. The machine still needs stable alignment, safe clearance, and resistance to rocking. Real installations often have several mounts, so load sharing matters.
If one mount carries more weight because of an uneven floor or poor adjustment, its deflection and natural frequency differ from the others. Measurements should include vibration frequency, amplitude, mounting direction, and operating condition. Comparing measurements before and after installation helps reveal whether the problem is resonance, imbalance, misalignment, loose foundations, or an unsuitable isolator.